1. Particle Physics
Michaelmas Term 2011
Prof Mark Thomson
Handout 10 : Leptonic Weak Interactions and Neutrino Deep Inelastic Scattering
Prof. M.A. Thomson
Michaelmas 2011

2. Aside : Neutrino Flavours
Recent experiments (see Handout 11) neutrinos have mass (albeit very small)
The textbook neutrino states, , are not the fundamental particles;
these are
Concepts like “electron number” conservation are now known not to hold.
So what are ?
Never directly observe neutrinos – can only detect them by their weak interactions.
Hence by definition is the neutrino state produced along with an electron.
Similarly, charged current weak interactions of the state produce an electron
= weak eigenstates n u d p ne e+ W e- ?
Unless dealing with very large distances: the neutrino produced with a positron
will interact to produce an electron. For the discussion of the weak interaction
continue to use as if they were the fundamental particle states.
Prof. M.A. Thomson
Michaelmas 2011

3. Muon Decay and Lepton Universality
The leptonic charged current (W±) interaction vertices are:
Consider muon decay:
It is straight-forward to write down the matrix element
Note: for lepton decay so propagator is a constant
i.e. in limit of Fermi theory
Its evaluation and subsequent treatment of a three-body decay is rather tricky
(and not particularly interesting). Here will simply quote the result
Prof. M.A. Thomson
Michaelmas 2011

4. The muon to electron rate with
Similarly for tau to electron
However, the tau can decay to a number of final states:
Recall total width (total transition rate) is the sum of the partial widths
Can relate partial decay width to total decay width and therefore lifetime:
Therefore predict
Prof. M.A. Thomson
Michaelmas 2011

5. Charged Current Lepton Universality
All these quantities are precisely measured:
Similarly by comparing and
Demonstrates the weak charged current is the same for all leptonic vertices
Charged Current Lepton Universality
Prof. M.A. Thomson
Michaelmas 2011

6. Neutrino Scattering Neutrino Beams: +
In handout 6 considered electron-proton Deep Inelastic Scattering where
a virtual photon is used to probe nucleon structure
Can also consider the weak interaction equivalent: Neutrino Deep Inelastic
Scattering where a virtual W-boson probes the structure of nucleons
additional information about parton structure functions
+ provides a good example of calculations of weak interaction cross sections.
Neutrino Beams:
Smash high energy protons into a fixed target
hadrons
Focus positive pions/kaons
Allow them to decay +
Gives a beam of “collimated”
Focus negative pions/kaons to give beam of
Proton beam
target
Magnetic
focussing
Decay tunnel
Absorber = rock
Neutrino
beam
Prof. M.A. Thomson
Michaelmas 2011

7. Neutrino-Quark Scattering
For proton Deep Inelastic Scattering the underlying process is p X q
In the limit the W-boson propagator is
The Feynman rules give:
Evaluate the matrix element in the extreme relativistic limit where the
muon and quark masses can be neglected
Prof. M.A. Thomson
Michaelmas 2011

8. where the matrix element is non-zero is
In this limit the helicity states are equivalent to the chiral states and
for
and
Since the weak interaction “conserves the helicity”, the only helicity combination
where the matrix element is non-zero is
NOTE: we could have written this down straight away as in the ultra-relativistic
limit only LH helicity particle states participate in the weak interaction.
Consider the scattering in the C.o.M frame
Prof. M.A. Thomson
Michaelmas 2011

9. Evaluation of Neutrino-Quark Scattering ME
Go through the calculation in gory detail (fortunately only one helicity combination)
In the CMS frame, neglecting particle masses:
Dealing with LH helicity particle spinors. From handout 3 (p.80), for a
massless particle travelling in direction :
Here
giving:
Prof. M.A. Thomson
Michaelmas 2011

10. need to evaluate two terms of form
To calculate
need to evaluate two terms of form
Using
Prof. M.A. Thomson
Michaelmas 2011

11. Note the Matrix Element is isotropic
we could have anticipated this since the
helicity combination (spins anti-parallel)
has
no preferred polar angle
As before need to sum over all possible spin states and average over
all possible initial state spin states. Here only one possible spin combination
(LLLL) and only 2 possible initial state combinations (the neutrino is always
produced in a LH helicity state)
The factor of a half arises because
half of the time the quark will be in
a RH states and won’t participate in
the charged current Weak interaction
From handout 1, in the extreme relativistic limit, the cross section for any
22 body scattering process is
Prof. M.A. Thomson
Michaelmas 2011

12. (1) using Integrating this isotropic distribution over
cross section is a Lorentz invariant quantity so this is valid in any frame
Prof. M.A. Thomson
Michaelmas 2011

13. Antineutrino-Quark Scattering
In the ultra-relativistic limit, the charged-current
interaction matrix element is:
In the extreme relativistic limit only LH Helicity particles and RH Helicity anti-
particles participate in the charged current weak interaction:
In terms of the particle spins it can be seen that the interaction occurs in a
total angular momentum 1 state
Prof. M.A. Thomson
Michaelmas 2011

14. The factor can be understood
In a similar manner to the neutrino-quark scattering calculation obtain:
The factor can be understood
in terms of the overlap of the initial and final
angular momentum wave-functions
Similarly to the neutrino-quark scattering calculation obtain:
Integrating over solid angle:
This is a factor three smaller than the neutrino quark cross-section
Prof. M.A. Thomson
Michaelmas 2011

15. (Anti)neutrino-(Anti)quark Scattering
Non-zero anti-quark component to the nucleon also consider scattering from
Cross-sections can be obtained immediately by comparing with quark scattering
and remembering to only include LH particles and RH anti-particles
Prof. M.A. Thomson
Michaelmas 2011

16. Differential Cross Section ds/dy
Derived differential neutrino scattering cross sections in C.o.M frame, can convert
to Lorentz invariant form
As for DIS use Lorentz invariant
In relativistic limit y can be expressed in terms
of the C.o.M. scattering angle
In lab. frame
Convert from using
Already calculated (1)
Hence:
Prof. M.A. Thomson
Michaelmas 2011

17. For comparison, the electro-magnetic cross section is:
and
becomes
from
and hence
For comparison, the electro-magnetic cross section is:
QED
DIFFERENCES:
Interaction
+propagator
Helicity
Structure
WEAK
Prof. M.A. Thomson
Michaelmas 2011

18. Parton Model For Neutrino Deep Inelastic ScatteringX q p X q
Scattering from a proton
with structure functions
Scattering from a point-like
quark within the proton
Neutrino-proton scattering can occur via scattering from a down-quark or
from an anti-up quark
In the parton model, number of down quarks within the proton in the
momentum fraction range is Their contribution to
the neutrino scattering cross-section is obtained by multiplying by the
cross-section derived previously
where is the centre-of-mass energy of the
Prof. M.A. Thomson
Michaelmas 2011

19. Similarly for the contribution
Summing the two contributions and using
The anti-neutrino proton differential cross section can be obtained in the
same manner:
For (anti)neutrino – neutron scattering:
Prof. M.A. Thomson
Michaelmas 2011

20. of proton/neutron scattering cross sections
As before, define neutron distributions functions in terms of those of the proton
(2)
(3)
(4)
(5)
Because neutrino cross sections are very small, need massive detectors.
These are usually made of Iron, hence, experimentally measure a combination
of proton/neutron scattering cross sections
Prof. M.A. Thomson
Michaelmas 2011

21. cross section per nucleon:
For an isoscalar target (i.e. equal numbers of protons and neutrons), the mean
cross section per nucleon:
Integrate over momentum fraction x
(6)
where and are the total momentum fractions carried by the quarks and
by the anti-quarks within a nucleon
Similarly
(7)
Prof. M.A. Thomson
Michaelmas 2011

22. e.g. CDHS Experiment (CERN 1976-1984)
1250 tons
Magnetized iron modules
Separated by drift chambers
Study Neutrino Deep
Inelastic Scattering N X
Experimental Signature:
Prof. M.A. Thomson
Michaelmas 2011

23. from curvature in B-field
Example Event:
Energy Deposited
Position
Hadronic
shower (X)
Muon
Measure energy of
Measure muon momentum
from curvature in B-field
For each event can determine neutrino energy and y !
Prof. M.A. Thomson
Michaelmas 2011

24. Measured y Distribution
u+d nN n
J. de Groot et al., Z.Phys. C1 (1979) 143
CDHS measured y distribution
Shapes can be understood in
terms of (anti)neutrino –
(anti)quark scattering
Prof. M.A. Thomson
Michaelmas 2011

25. Measured Total Cross Sections
Integrating the expressions for (equations (6) and (7))
DIS cross section lab. frame neutrino energy
Measure cross sections can be used to determine fraction of protons momentum
carried by quarks, , and fraction carried by anti-quarks,
Find:
~50% of momentum carried by gluons
(which don’t interact with virtual W boson)
If no anti-quarks in nucleons expect
Including anti-quarks
Prof. M.A. Thomson
Michaelmas 2011

26. F.J. Hasert et al., Phys. Lett. 46B (1973) 121
Weak Neutral Current
Neutrinos also interact via the Neutral Current. First observed in the Gargamelle
bubble chamber in Interaction of muon neutrinos produce a final state muon
F.J. Hasert et al., Phys. Lett. 46B (1973) 121
F.J. Hasert et al., Phys. Lett. 46B (1973) 138 nm
Cannot be due to W exchange - first evidence for Z boson
Prof. M.A. Thomson
Michaelmas 2011

27. Summary suppressed by factor compared with
Derived neutrino/anti-neutrino – quark/anti-quark weak charged current (CC)
interaction cross sections
Neutrino – nucleon scattering yields extra information about parton
distributions functions:
couples to and ; couples to and
investigate flavour content of nucleon
can measure anti-quark content of nucleon
suppressed by factor compared with
suppressed by factor compared with
Further aspects of neutrino deep-inelastic scattering (expressed in general
structure functions) are covered in Appendix II
Finally observe that neutrinos interact via weak neutral currents (NC)
Prof. M.A. Thomson
Michaelmas 2011

28. Appendix I For the adjoint spinors consider
Using the fact that and anti-commute can rewrite ME:
for
and
Prof. M.A. Thomson
Michaelmas 2011

29. Appendix II: Deep-Inelastic Neutrino Scatteringq p X q
Two steps:
First write down most general cross section in terms of structure functions
Then evaluate expressions in the quark-parton model
QED Revisited
In the limit the most general electro-magnetic deep-inelastic
cross section (from single photon exchange) can be written (Eq. 2 of handout 6)
For neutrino scattering typically measure the energy of the produced muon
and differential cross-sections expressed in terms of
Using
Prof. M.A. Thomson
Michaelmas 2011

30. term to allow for parity violation. New structure function
In the limit the general Electro-magnetic DIS cross section can be written
NOTE: This is the most general Lorentz Invariant parity conserving expression
For neutrino DIS parity is violated and the general expression includes an additional
term to allow for parity violation. New structure function
For anti-neutrino scattering new structure function enters with opposite sign
Similarly for neutrino-neutron scattering
Prof. M.A. Thomson
Michaelmas 2011

31. Neutrino Interaction Structure Functions
In terms of the parton distribution functions we found (2) :
Compare coefficients of y with the general Lorentz Invariant form (p.321) and
assume Bjorken scaling, i.e.
Re-writing (2)
and equating powers of y
gives:
Prof. M.A. Thomson
Michaelmas 2011

32. NOTE: again we get the Callan-Gross relation
No surprise, underlying process is scattering from point-like spin-1/2 quarks
Substituting back in to expression for differential cross section:
Experimentally measure and from y distributions at fixed x
Different y dependencies (from different rest frame angular distributions)
allow contributions from the two structure functions to be measured
“Measurement”
Then use and
Determine and separately
Prof. M.A. Thomson
Michaelmas 2011

33. Neutrino experiments require large detectors (often iron) i. e
Neutrino experiments require large detectors (often iron) i.e. isoscalar target
For electron – nucleon scattering:
For an isoscalar target
Note that the factor and by comparing neutrino to
electron scattering structure functions measure the sum of quark charges
Experiment:  0.02
Prof. M.A. Thomson
Michaelmas 2011

34. Measurements of F2(x) and F3(x)
CDHS Experiment
H. Abramowicz et al., Z.Phys. C17 (1983) 283 nN
Difference in neutrino structure
functions measures anti-quark
(sea) parton distribution functions
QED DIS
Sea dominates so expect xF3
to go to zero as q(x) = q(x)
Sea contribution goes to zero
Prof. M.A. Thomson
Michaelmas 2011

35. Valence Contribution “Gross – Llewellyn-Smith sum rule”
Separate parton density functions into sea and valence components
Area under measured function gives a measurement of the total
number of valence quarks in a nucleon !
“Gross – Llewellyn-Smith sum rule”
expect
Experiment: 3.0±0.2
Note: and anti-neutrino
structure functions contain same pdf information
Prof. M.A. Thomson
Michaelmas 2011